Nonlinear Control Sy.. - Free

3.12. EXERCISES 103
(a) Find all of its equilibrium points.
(b) For each equilibrium point xe different from zero, perform a change of variables
y = x - xef and show that the resulting system y = g(y) has an equilibrium point
at the origin.
(3.2) Given the systems (i) and (ii) below, proceed as follows:
(a) Find all of their equilibrium points.
(b) Find the linear approximation about each equilibrium point, find the eigenvalues
of the resulting A matrix and classify the stability of each equilibrium point.
(c) Using a computer package, construct the phase portrait of each nonlinear system
and discuss the qualitative behavior of the system. Make sure that your analysis
contains information about all the equilibrium points of these systems.
(d) Using the same computer package used in part (c), construct the phase portrait
of each linear approximation found in (c) and compare it with the results in part
(c). What can you conclude about the "accuracy" of the linear approximations
as the trajectories deviate from the equilibrium points.
(Z)
J
ll x2 = x, - 2 tan 1(x1 + x2) , (ii) I x2
(3.3) Consider the magnetic suspension system of Section 1.9.1:
X z = g -
X3 =
M I 2m(1 +µx,)2
1+µx1 Aµ 1
A
[_+ (1 + px,)2x2x3 + vj
23 X2
-x, +x2(1 - 3x1 - 2x2)
(a) Find the input voltage v = vo necessary to keep the ball at an arbitrary position
y = yo (and so x, = yo). Find the equilibrium point xe = [xel , xe2, xe3]
corresponding to this input.
(b) Find the linear approximation about this equilibrium point and analyze its stability.
(3.4) For each of the following systems, study the stability of the equilibrium point at the
origin:
(Z)
f 1 = -X1 - x2x2
22 = -X2 - X1X2
,
(ii) 1
22
-X + x,x2
2 3 -xlx2 - x2
l